Abstract

Nowadays, the boundary crossing problem of diffusion processes is of interest to both mathematicians and statisticians. In this thesis, we review the literature on the first passage time problem for both one-dimensional and two-dimensional diffusion processes. Then we investigate the statistical inference problem about unknown parameters of the Cox-Ingersoll-Ross model based on discretely observed first passage times. We are able to determine the identifiable parameter set, discuss the tail property of the density function in a neighborhood of the true parameter, and propose a conditional version of maximum likelihood estimation. We also list future work, including extensions of this problem to a general one-dimensional time homogeneous diffusion process, and to some special two-dimensional diffusion processes.